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I'm trying to derive the mean thermal wavelength from the Maxwell distribution:

[tex]M(v) = 4\pi\left(\frac{m}{2\pi k_BT}\right)^{3/2}v^2e^{-\frac{mv^2}{2k_BT}} = 4\pi\left(\frac{a}{\pi}\right)^{3/2}v^2e^{-av^2}[/tex]

With [tex]a = \frac{m}{2k_BT}[/tex] introduced for convenience. Since [tex]\lambda = \frac{h}{mv}[/tex], I figured an expression for the thermal wavelength would be

[tex]\lambda_T = \frac{h}{m}\int_0^\infty\frac{M(v)}{v}dv[/tex]

Problem is, when I do this, I end up with

[tex]\lambda_T = 2\sqrt{\frac{h^2}{2\pi m k_BT}}[/tex]

This is exactly twice the accepted value. Where did I go wrong?

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# Derivation of Mean Thermal Wavelength

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